Response to Discretising and validating Keyfitz' entropy for any demographic classification

Abstract

In recent years, demographers have parsed variation in survivorship into two distinct components: the shape and the pace of ageing. The pace of ageing is defined by measures of longevity such as mean life expectancy or maximum longevity, whereas measures of the shape of ageing attempt to classify different shapes of the survivorship curve. We recently published a paper pointing out that a commonly used discretization of a shape measure, Keyfitz' entropy, does not correctly classify survivorship curves into negatively and positively senescing curves (de Vries et al., 2023). In that paper, we also suggested an alternative, accurate discrete-time version of Keyfitz' entropy. de Vries et al. (2023) ended with two open questions, both of which have been answered by Giaimo (2024) now: (1) Can a discrete-time entropy measure of survivorship be generalized beyond age-based population models? Giaimo (2024) introduce a new formula that achieves this. (2) Will a discretization of Keyfitz' derivation of his measure as the elasticity of lifespan to a uniform change in mortality lead to the same formula as a discretization of Keyfitz' result? Giaimo (2024) answers: no. Here, we briefly discuss the implications of the results obtained by Giaimo (2024), and the implementation of his new formula into the Rpackage Rage, which is widely used for comparative demographic studies. We showcase the strength of the new method by reanalysing a comparison of Keyfitz' entropy to another shape measure. We find that the comparison is significantly altered by using Giaimo's new Keyfitz' formula. This example strengthens Giaimo's (2004) words of warning in approaching discretizations with attentiveness.